integral of cos^3(X)sin^2(X)DX

Lösung

\int cos^{3} (X) sin^{2} (X) D X d X

D (\frac{1}{3} X sin^{3} (X) - \frac{1}{5} X sin^{5} (X) + \frac{2}{15} cos (X) + \frac{1}{45} cos^{3} (X) - \frac{1}{25} cos^{5} (X)) + C

Schritte zur Lösung

\int cos^{3} (X) sin^{2} (X) D X d X

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= D \cdot \int cos^{3} (X) sin^{2} (X) X d X

Wende die partielle Integration an

= D (X (\frac{1}{3} sin^{3} (X) - \frac{1}{5} sin^{5} (X)) - \int \frac{1}{3} sin^{3} (X) - \frac{1}{5} sin^{5} (X) d X)

\int \frac{1}{3} sin^{3} (X) - \frac{1}{5} sin^{5} (X) d X = - \frac{2}{15} cos (X) + \frac{1}{9} cos^{3} (X) - \frac{2}{15} cos^{3} (X) + \frac{1}{25} cos^{5} (X)

= D (X (\frac{1}{3} sin^{3} (X) - \frac{1}{5} sin^{5} (X)) - (- \frac{2}{15} cos (X) + \frac{1}{9} cos^{3} (X) - \frac{2}{15} cos^{3} (X) + \frac{1}{25} cos^{5} (X)))

Vereinfache

D (X (\frac{1}{3} sin^{3} (X) - \frac{1}{5} sin^{5} (X)) - (- \frac{2}{15} cos (X) + \frac{1}{9} cos^{3} (X) - \frac{2}{15} cos^{3} (X) + \frac{1}{25} cos^{5} (X))) : D (\frac{1}{3} X sin^{3} (X) - \frac{1}{5} X sin^{5} (X) + \frac{2}{15} cos (X) + \frac{1}{45} cos^{3} (X) - \frac{1}{25} cos^{5} (X))

= D (\frac{1}{3} X sin^{3} (X) - \frac{1}{5} X sin^{5} (X) + \frac{2}{15} cos (X) + \frac{1}{45} cos^{3} (X) - \frac{1}{25} cos^{5} (X))

Füge eine Konstante zur Lösung hinzu

= D (\frac{1}{3} X sin^{3} (X) - \frac{1}{5} X sin^{5} (X) + \frac{2}{15} cos (X) + \frac{1}{45} cos^{3} (X) - \frac{1}{25} cos^{5} (X)) + C

\int 2 cos (x) - 2 sin (x) d x

\int - 2 x cos (8 x) d x

in v erse l a pl a ce \frac{5}{s ( 4 s ^{2} + 4 s + 1 )}

x \to 3 lim (- x 6 x - 8)

\int_{- \infty}^{1} 3^{x} d x